I think there is information about the mine owner, Jacob Walton, in the local archives.  If you write to:  The Archivist, c/o Local Links, Town Hall, Alston, Cumbria, your letter will be passed on.  If you ring the Records Office in Carlisle they, also, had a box, I think it was called WAL/1, of Walton records.  Hope you get lucky!
Regards, Vera